A$2.(12$ pts$)$ Take $f$ to be the polynomial function that is given by the formula

$f(x)=x^3-6x^2+6x-5,$

and $g$ to be the function given by this sketch:

The line sketched in the above figure is tangent to $g$ at its point of intersection.

$($a$)$ Write $f$ as the sum of a line and an error term to identify an equation for the line $L$ that is tangent to $f$ at $(1,-4).$

$($b$)$ Determine the order of intersection of $L$ with $f$ at $(1,-4).$

$($c$)$ Determine $(2f+5g{)}^{\text{'}}(1).$

$($d$)$ Determine $(fg{)}^{\text{'}}(1).$